In [1]} an easy proof of the General Néron Desingularization is given in the case when \(\dim A=0\) , \(\dim A^{\prime }=1\)  and \(A\) , \(A^{\prime }\)  have the same residue field. Here
it is our goal to provide some algorithms in this direction. We start presenting an algorithm to construct the Néron
Desingularization in the case when \(A\) , \(A^{\prime }\)  are DVR. In Section 1 we recall Néron's proof in the idea presented by Artin [2]} in a special case (see also [3]}, [4]}). The present proof is taken from [5]} and we give it here for the sake of completeness.
In Sections 2, 3 we present shortly a constructive proof and give an algorithm to construct Néron Desingularization in the easier case when \(A\) , \(A^{\prime }\)  have the same residue field as in [6]}. We do not implement our algorithm. The implementation done in [7]} gives a hard General Néron Desingularization because it is constructed in a general case, that is when \(A\) , \(A^{\prime }\)  are Noetherian local domains of one dimension and with possible different residue fields and so involving some extra equations and new variables.
In the last sections we present an algorithm to construct the easy case from [1]} of the General Néron Desingularization.
